Consider the curve in the plane originally parameterized by as for

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

Consider the curve CC in the plane originally parameterized by θ\theta as r(θ)=⟨θcos⁡θ, θsin⁡θ⟩\mathbf r(\theta)=\langle \theta\cos\theta,\,\theta\sin\theta\rangle for 0≤θ≤2π0\le \theta\le 2\pi. The curve is re-parameterized by tt where θ=t2\theta=t^{2}. (a) Write the position vector of the curve in terms of the new parameter tt, and state the corresponding interval for tt. (b) Derive an explicit expression for the speed of the particle as a function of tt under the new parameterization. (c) Determine the exact length of the curve over the entire interval of tt found in part (a). (d) Does the re-parameterization alter the geometric length of the curve? Provide a brief justification based on your work in part (c).

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